13.2 Angles of Rotation

Holt Algebra 2

13-2 Angles of Rotation

Warm Up

Find the measure of the supplement for each given angle.

1. 150° 2. 120° 3. 135° 4. 95°

5. Find the value of the sine, cosine, and tangent functions for θ.

30° 60°

45° 85°

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Draw angles in standard position.

Determine the values of the trigonometric functions for an angle in standard position.

Objectives

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standard positioninitial sideterminal sideangle of rotationcoterminal anglereference angle

Vocabulary

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In Lesson 13-1, you investigated trigonometric functions by using acute angles in right triangles. The trigonometric functions can also be evaluated for other types of angles.

An angle is in standard position when its vertex is at the origin and one ray is on the positive x-axis. The initial side of the angle is the ray on the x-axis. The other ray is called the terminal side of the angle.

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An angle of rotation is formed by rotating the terminal side and keeping the initial side in place. If the terminal side is rotated counterclockwise, the angle of rotation is positive. If the terminal side is rotated clockwise, the angle of rotation is negative. The terminal side can be rotated more than 360°.

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A 360° rotation is a complete rotation. A 180° rotation is one-half of a complete rotation.

Remember!

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Example 1: Drawing Angles in Standard Position

Draw an angle with the given measure in standard position.

A. 320°

Rotate the terminal side 320° counterclockwise.

B. –110° C. 990°

Rotate theterminal side –110° clockwise.

Rotate the terminal side 990° counterclockwise.

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Rotate the terminalside 210° counter-clockwise.

Rotate the terminalside 1020° counter-clockwise.

Rotate the terminalside 300° clockwise.

Check It Out! Example 1

Draw an angle with the given measure in standard position.

A. 210° B. 1020° C. –300°

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Coterminal angles are angles in standard position with the same terminal side. For example, angles measuring 120° and – 240° are coterminal.

There are infinitely many coterminal angles. One way to find the measure of an angle that is coterminal with an angle θ is to add or subtract integer multiples of 360°.

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Example 2A: Finding Coterminal Angles

Find the measures of a positive angle and a negative angle that are coterminal with each given angle.

= 65°

65° + 360° = 425°

65° – 360° = –295°

Add 360° to find a positive coterminal angle.

Subtract 360° to find a negative coterminal angle.

Angles that measure 425° and –295° are coterminal with a 65° angle.

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Example 2B: Finding Coterminal Angles


= 410°

410° – 360° = 50°

410° – 2(360°) = –310°

Subtract 360° to find a positive coterminal angle.

Subtract a multiple of 360° to find a negative coterminal angle.


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Check It Out! Example 2a


= 88°

88° + 360° = 448°

88° – 360° = –272°



Angles that measure 448° and –272° are coterminal with an 88° angle.

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Check It Out! Example 2b


= 500°

500° + 360° = 860°

500° – 2(360°) = –220°


Subtract a multiple of 360° to find a negative coterminal angle.


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Check It Out! Example 2c


= –120°

–120° + 360° = 240°

–120° – 360° = –480°



Angles that measure 240° and –480° are coterminal with a –120° angle.

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Check It Out! Example 2a


A. = 88°

B. = 500°

C. = –120°

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For an angle θ in standard position, the reference angle is the positive acute angle formed by the terminal side of θ and the x-axis.

In Lesson 13-3, you will learn how to use reference angles to find trigonometric values of angles measuring greater than 90° or less than 0°.

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Example 3: Finding Reference Angles

Find the measure of the reference angle for each given angle.

B. = –105°A. = 135°

The measure of the reference angle is 45°.


–105°

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Example 3: Finding Reference Angles


C. = 325°


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105°

The measure of thereference angle is 75°

–115°


a. = 105° b. = –115°

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310°


c. = 310°

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a. = 105° b. = –115°

c. = 310°

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To determine the value of the trigonometric functions for an angle θ in standard position, begin by selecting a point P with coordinates (x, y) on the terminal side of the angle. The distance r from point P to the origin is given by .

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Example 4: Finding Values of Trigonometric Functions

P (–6, 9) is a point on the terminal side of in standard position. Find the exact value of the six trigonometric functions for θ.

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Example 4: Finding Values of Trigonometric Functions

P (–6, 9) is a point on the terminal side of in standard position. Find the exact value of the six trigonometric functions for θ.

Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

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Step 2 Find sin θ, cos θ, and tan θ.

Example 4 Continued

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Example 4 Continued

Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.

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Because r is a distance, its value is always positive, regardless of the sign of x and y.

Helpful Hint

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P(–3, 6) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ.

y

x

P(–3, 6)

θ

Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

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Check It Out! Example 4 Continued

Step 2 Find sin θ, cos θ, and tan θ.

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Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.

Check It Out! Example 4 Continued

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P(–3, 6) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ.

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Lesson Quiz: Part I

Draw and angle in standard position with the given measure.

1. 210° 2. –160°

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Lesson Quiz: Part II

3. θ = 290° 4. θ = –195°70° 15°

5. P(1, –1) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ.

13.2 Angles of Rotation

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